Convergence Of Numerical Methods For Some Classes Of Stiff Problems

The main results in this paper are the following:(1) We study the B-convergence of additive Runge-Kutta methods (in- cluding multiple time-step methods and fractional step methods) for a class of nonlinear dissipative stiff problems, present that the order of optimal B- convergence of algebraically stable and ANS-stable additive Runge-Kutta methods is equal at least to the stage order, and provide a necessary and su- fficient condition for order of optimal B-convergence one higher than the stage order. These extend and develop the relevant results of Runge-Kutta methods given by Burrage etc. in 1987 for this class of problems, and also partly extend the corresponding results given by Aiguo Xiao in 1992.(2) Backward Differentiation Formulas (BDFs) are very important and efficient methods for the solution of stiff problems. In this paper, based on a new approach, we study the error properties of backward differentiation for- mulas for a class of semi-linear stiff problems, whose stiffness is contained in the constant or variable coefficients linear part, and present the quantita- tive convergence results for global error. These extend and develop the cor- responding results of backward differentiation formulas for constant or va- riable coefficients linear stiff problems given by Kirlinger etc. in 2001.